If I have a topological space $(E,\tau)$ that is metrizable for a metric $d$, and then I look at the set of all open balls in this metric $d$, will this always form a topology? If yes, will this topology be the same $\tau$ ?
- Yes, the topology induced by the open balls for the metric $d$ will be a topology.
I want to specify that the topology induced by the open balls is not the set of open balls, but the set of union of open balls.
- But no, there is no reason that this topology would be the same as $\tau$.
Take for instance $E=\mathbb R$, $d=\vert .\vert$ and $\tau=\mathfrak P(\mathbb R)$ the discrete topology where every subset is open.
You have have two different topologies.
The answer for your second question is yes if you consider that metrizable means that the metric need to be deduced from the topology.